Witten index for weak supersymmetric systems: invariance under deformations

TL;DR

This paper investigates the invariance of the Witten index in weak supersymmetric systems on curved spaces, showing it remains unchanged under certain deformations despite not being an integer.

Contribution

It provides a general proof of the Witten index's invariance in weak supersymmetric systems and illustrates this with simple quantum mechanical models.

Findings

Witten index is a $eta$-dependent function in weak SUSY systems.

The index remains invariant under deformations preserving the SUSY algebra.

Explicit examples demonstrate the invariance in simple quantum systems.

Abstract

When a $4D$ supersymmetric theory is placed on $S^3 \times \mathbb{R}$, the supersymmetric algebra is necessarily modified to $su(2|1)$ and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index Tr$\{(-1)^F e^{-\beta H}\}$ is no longer an integer number, but a $\beta$-dependent function. However, this function stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.